Lover of math. Bad at drawing.

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The Number Line: A Journey

Everybody thinks they grew up in the most boring neighborhood on the whole number line. But trust me: you’d rather spend seven eternities in your home than a Saturday night in mine.

I don’t need to describe my home interval; you know it already. Neither terribly close to an integer, nor terribly far from one. No famous constants for miles around. No crossroads, no bustling port, no frontier town: just an anonymous suburb of an unseen city, a faceless stretch on the long gray road that runs from one infinity to the other.

Nothing to do? Understatement. There was nothing to think, nothing to feel, nothing to say. I gathered stories wherever I could, scavenged for tales of distant realms where something, anything, was happening.

Somewhere, a trillion covert sequences spiraled towards pi.

Somewhere, a mirror reflected the negative image of every number I knew.

Somewhere, the integers climbed beyond trillions in an Icarus flight to infinity.

Somewhere, somewhere, somewhere…

Adulthood came without sentiment or fanfare, like the bell at the end of third period. I simply gathered my things and made for the door. I’d like to say that I hugged my parents a half-decent goodbye, but the truth is that I left without a word.

I traveled first to the hub, the nexus, the galactic center around which everything else swirled.

I went to zero.

Myth and magic sucked me in. For a while I simply staggered from site to site, a star-struck tourist.

I walked the unit interval, gawking shamelessly, unable to reconcile this humble stretch of number line (“it looks just like home!”) with the thousand legendary proofs and immortal computations that had taken place right here.

I rode one famous geometric sequence after the next: 1/2, 1/4, 1/8, 1/16… then 1/3, 1/9, 1/27, 1/81… and so on, and so on, each trail starting out colorfully distinct, but growing more and more familiar as it converged on that Grand Central Station to which every rail led.

And at last I stood, crowded alongside countless pilgrims, on the cusp of zero itself.

I don’t know what I expected to feel. Joy? Awe? Fulfillment? I felt none of these, only a numb giddiness that wore off the moment I stepped away. Beneath the foamy glamour of Zero’s city, what was there, really?

A null, a void.

I learned that day, as the cheap thrill and soft buzz of celebrity faded, that Zero is just nothingness with a good PR team.

And so I moved on, and with a single step crossed into the country of the negatives.

I staggered, adjusting to the eerie anti-light, feeling as queasy and free as if I’d flipped the switch on gravity.

In the realm of the negatives, nothing was different, and nothing the same.

I understood the locals, and they me, but we could not speak a word of each other’s languages. We could only marvel wide-eyed at the strange customs and backwards habits that the other displayed.

I had never questioned what it meant to add or subtract. But their cheery and utterly opposite practices forced me to reconsider, to seek the deeper logic that could reconcile my home with theirs.

Time passed, their clock hands circling in a motion I’d have once called “wrong,” but increasingly understood as simply a different sort of “right.”

Eventually, though, I grew weary and ready to leave.

At the time, I attributed my frustration to the drab familiarity of the place: I had come for adventure, and found a country as provincial and set in its ways as the one I left behind! Looking back, though, I suspect I felt something more like homesickness. Every inverted tradition was a tiny, sharp reminder that I was in alien territory, far from those I knew and who knew me.

But who understands their own urges and needs as they occur? Not I, certainly. So, instead of making for home, I fled chasing a rumor of a truly different land, far more alien than the negatives:

I sought a realm that lay beyond the number line altogether.

It sounded absurd. Indeed, it was absurd. But if the gods of number have one lesson to teach us, it’s that absurdity and truth are not always opposed.

In describing this episode of my life’s journey, I risk slipping into tedious psychedelia and hippie platitudes. But here it goes.

I saw another dimension.

I learned that our linear world is just a tiny sliver of a cosmos vaster than imagination.

In the complex plane, you’re free to stretch in undreamed directions, to flex undiscovered muscles.

It was far out, man.

I would slide up the imaginary axis and then veer away, dipping back down, crossing momentarily through the number line (to the bewilderment of the locals), then emerge exultant on the far side.

I don’t know how long I spent in that waking dream.

I only know that eventually I found myself drifting along parallel to the real number line, unthinking, unsure of my destination, until suddenly I was staring from the outside at a very familiar interval.

Home.

I watched my parents going about their day. I could see their wrinkles, hear their soft murmurs, anticipate their smiles and sighs.

I was increments away, mere epsilons.

But they could not sense me; I was elsewhere, just beyond their reality. I lay outside the easy solidity of their lives. I was something complex now, part imaginary.

Gazing at my home interval I felt a total and utter longing, like emptiness dreaming of substance.

But I couldn’t admit it to myself. Not yet. In my mind, returning was weakness, and destiny lay always ahead.

Some say the whole complex plane is a druggie sham, a mirage, a persistent and collective hallucination. I’m not so sure – when I was there I saw things, felt things, too profound and too real to be written off. But real or not – delusion or dream or dimension unexplored – I knew even then that the complex realm was not for me.

So I rejoined the real number line, on the far side of home, and transferred my sights to the horizon.

I then began my pursuit of the Promised Land, the terrestrial heaven just out of reach, to which every soul aspires.

I set out to find infinity.

Growing up, you hear a lot about infinity, and none of it coheres. Infinity is the largest number. Infinity isn’t a number at all. All infinities are identical. Each is bigger than the last.

I only knew one thing for sure: which direction to head. And so I did.

I climbed past millions and billions.

I climbed past quadrillions and googols.

The numbers’ names grew longer. Eventually, I passed an integer so enormous that it could not be described in fewer than a hundred syllables. (At least, I thought I did, though when I looked closer, I realized “the first number that cannot be described in fewer than a hundred syllables” is only 20 syllables.)

Eventually, I tread upon unholy cosmic numbers whose unutterable names would take a trillion trillion years to pronounce, numbers so large they made galaxies quaint.

Time passed in eons.

I journeyed on.

And still I did not reach infinity.

One day I stopped. I gazed into the distance, straining to see the end. But I glimpsed nothing, only that long gray road vanishing into the haze.

I realized it then: Infinity isn’t a place. It’s simply a name we give to journeys that never end.

And I also realized – late, so late, though never too late – that I didn’t want such a journey.

I didn’t aspire to be infinite.

I wanted to go home.

And so, with a tiny pivot, I tore my eyes from the boundless horizon and took my first step back towards home.

When I arrived, my parents were facing towards zero, gazing in the direction I’d left. I expected them to be surprised, confused, when I arrived from the opposite side. But they exhibited only an explosive and joyous relief.

Without sentiment or fanfare, I settled back into home life, staked out my own little patch along the interval.

And one day, not long after, I gazed down at the unremarkable stretch of number line below me. For the first time, I really studied it.

For the first time, I saw.

I saw, within my interval, a dozen smaller ones; and within each of those, a dozen more; and within each of those, more and more, so on and so on, a fractal intricacy I’d somehow never glimpsed.

I saw the rational numbers, densely packed as grains of sand, an infinite congregation of them in every tiny neighborhood.

I saw the irrational numbers, inexpressible as the ratio of whole numbers, fierce and alien as anything I’d seen on my journeys.

I’d always assumed most numbers belonged to that blander, more predictable family: the rationals. But looking now, it dawned on me that the irrationals were far vaster. The irrationals were as untouchable as ether yet as common as mud. Virtually the whole world was sewn from their shimmering and unknowable fabric, with the rationals set like rare jewels among them.

I gazed down at the continuum between my toes and realized that there was infinite beauty, infinite mystery, everywhere.

In every interval.

In every neighborhood.

Every patch of number line contained the intricate fingerprints of an impossible act of creation.

I glanced over at my parents, living out their day on this dull interval, no different from any other, and I smiled.

21 thoughts on “The Number Line: A Journey”

I love this! I wish I had something more clever to say, but I’m tired, so I’ll just say that I really enjoyed this. It’s like a Wizard of Oz story for numbers, has a science fictiony, dreamy, psychological, philosophical feel. You obviously took time with it.

We often only think of math as only a means to an end. A calculation of volume or something in science as a numerical formula that the natural world must follow. “You see, this is why we learn math”, someone might say. This however, misses the heart of mathematics. It is a wonder in its own right. It is its own art, but so many miss that. It is its own end through whatever means. Thank you for this journey. It is very well written. Very mysterious, but yet something I’ve pondered upon my entire life.

Reminds me of the journey to the land of quadrants where positive and negative numbers combine to create beautiful landscapes instead of doing battle trying to cancel each other out or throwing magic spells at each other to change their sign.